Infinite-dimensional-vector-valued function

Infinite-dimensional vector function refers to a function whose values lie in an infinite-dimensional vector space, such as a Hilbert space or a Banach space.

Such functions are applied in most sciences including physics.

Set ${\displaystyle f_{k}(t)=t/k^{2}}$ for every positive integer k and every real number t. Then values of the function

lie in the infinite-dimensional vector space X (or ${\displaystyle \mathbf {R} ^{\mathbf {N} }}$) of real-valued sequences. For example,

As a number of different topologies can be defined on the space X, we cannot talk about the derivative of f without first defining the topology of X or the concept of a limit in X.

Moreover, for any set A, there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of A (e.g., the space of functions ${\displaystyle A\rightarrow K}$ with finitely-many nonzero elements, where K is the desired field of scalars). Furthermore, the argument t could lie in any set instead of the set of real numbers.

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